Sequences and Series: Modelling

Sequences and Series: Modelling

Understanding Modelling of Sequences and Series

  • Modelling involves using a mathematical structure to represent real-world phenomena and then to predict or explain the phenomena.
  • In sequences and series, the modelling process may involve setting up a sequence to describe a situation, or fitting a sequence to data.
  • A series is the summed terms of a sequence, whereas the sequence is simply the list of numbers.

Arithmetic Sequences and Series

  • An arithmetic sequence is a sequence of numbers in which the difference of between any two successive members is constant.
  • The nth term of an arithmetic sequence can be given by a + (n-1)d, where a is the first term and d is the common difference.
  • The sum of an arithmetic series, S_n, of n terms can be calculated using the formula S_n = n/2(2a + (n - 1)d).

Geometric Sequences and Series

  • A geometric sequence is a sequence of numbers in which the ratio of any two successive members is constant. This constant is often called the common ratio.
  • The nth term of a geometric sequence can be given by ar^(n-1), where a is the first term and r is the common ratio.
  • The sum of a geometric series, S_n, of n terms can be calculated using the formula S_n = a(1 - r^n) / (1 - r), where r < 1.

Using Sequences and Series in Modelling

  • In modelling physical or biological phenomena, arithmetic and geometric sequences and series are often used to represent or approximate the phenomena.
  • The choice between arithmetic and geometric sequences and series often depends on the nature of the phenomena, such as whether the phenomena exhibit constant change or exponential change.

The Limit of a Series

  • The limit of a series is a value that the series approaches as the number of terms goes to infinity.
  • For a geometric series where r < 1, the series has a limit, given by a / (1 - r).

Fibonacci and Other Sequences

  • There are many sequences other than arithmetic and geometric sequences, such as the Fibonacci sequence, which can also be used in modelling.
  • Fibonacci sequence is defined by the recurrence relation f(n) = f(n - 1) + f(n - 2) with initial conditions f(0) = 0 and f(1) = 1.
  • In a similar way to arithmetic and geometric sequences, the properties of these sequences can be analysed, and the sequences can be used to model real-world phenomena.